Necessary and sufficient conditions for the bounds of the commutator of a Littlewood-Paley operator with fractional differentiation

Abstract

For $b\in L_{\mathrm{loc}}({\mathbb {R}}^n)$ and $0<\alpha <1$, we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely

$$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg |^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$

Here, we obtain the necessary and sufficient conditions for the function b to guarantee that $g_{\Omega ,\alpha ;b}$ is a bounded operator on $L^2({\mathbb {R}}^n)$. More precisely, if $\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}$ and $b\in I_{\alpha }(BMO)$, then $g_{\Omega ,\alpha ;b}$ is bounded on $L^2({\mathbb {R}}^n)$. Conversely, if $g_{\Omega ,\alpha ;b}$ is bounded on $L^2({\mathbb {R}}^n)$, then $b \in Lip_\alpha ({\mathbb {R}}^n)$ for $0<\alpha < 1$.